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Index theory of hypoelliptic operators on Carnot manifolds

Magnus Goffeng, Alexey Kuzmin

TL;DR

The work develops a geometric K-theory framework for the index problem of hypoelliptic (H-elliptic) operators on regular Carnot manifolds by localization to flat coadjoint orbits, and by constructing a bundle of flat orbit representations that realizes Morita-type equivalences. It extends Baum–van Erp's index theory from co-oriented contact manifolds to regular polycontact Carnot manifolds, providing explicit dualities between analytic and geometric K-homology and yielding concrete index formulas for Toeplitz-type and Hörmander-type operators. The approach combines refined representation theory of nilpotent Lie groups, fine stratifications of the unitary dual, and groupoid techniques (Connes–Thom, adiabatic groupoids) to produce computable K-theoretic expressions for indices, along with an overarching framework that handles noncompact base manifolds via a noncompact geometric K-homology model. The results include a broad spectrum of examples, concrete index formulas, and vanishing results, illustrating the method's power and clarifying the role of flat orbits and metaplectic corrections in the index problem on Carnot manifolds.

Abstract

We study the index theory of hypoelliptic operators on Carnot manifolds -- manifolds whose Lie algebra of vector fields is equipped with a filtration induced from sub-bundles of the tangent bundle. A Heisenberg pseudodifferential operator, elliptic in the calculus of van Erp-Yuncken, is hypoelliptic and Fredholm. Under some geometric conditions, we compute its Fredholm index by means of operator $K$-theory. These results extend the work of Baum-van Erp (Acta Mathematica '2014) for co-oriented contact manifolds to a methodology for solving this index problem geometrically on Carnot manifolds. Under the assumption that the Carnot manifold is regular, i.e. has isomorphic osculating Lie algebras in all fibres, and admits a flat coadjoint orbit, the methodology derived from Baum-van Erp's work is developed in full detail. In this case, we develope $K$-theoretical dualities computing the Fredholm index by means of geometric $K$-homology a la Baum-Douglas. The duality involves a Hilbert space bundle of flat orbit representations. Explicit solutions to the index problem for Toeplitz operators and operators of the form "$Δ_H+γT$" are computed in geometric $K$-homology, extending results of Boutet de Monvel and Baum-van Erp, respectively, from co-oriented contact manifolds to regular polycontact manifolds.

Index theory of hypoelliptic operators on Carnot manifolds

TL;DR

The work develops a geometric K-theory framework for the index problem of hypoelliptic (H-elliptic) operators on regular Carnot manifolds by localization to flat coadjoint orbits, and by constructing a bundle of flat orbit representations that realizes Morita-type equivalences. It extends Baum–van Erp's index theory from co-oriented contact manifolds to regular polycontact Carnot manifolds, providing explicit dualities between analytic and geometric K-homology and yielding concrete index formulas for Toeplitz-type and Hörmander-type operators. The approach combines refined representation theory of nilpotent Lie groups, fine stratifications of the unitary dual, and groupoid techniques (Connes–Thom, adiabatic groupoids) to produce computable K-theoretic expressions for indices, along with an overarching framework that handles noncompact base manifolds via a noncompact geometric K-homology model. The results include a broad spectrum of examples, concrete index formulas, and vanishing results, illustrating the method's power and clarifying the role of flat orbits and metaplectic corrections in the index problem on Carnot manifolds.

Abstract

We study the index theory of hypoelliptic operators on Carnot manifolds -- manifolds whose Lie algebra of vector fields is equipped with a filtration induced from sub-bundles of the tangent bundle. A Heisenberg pseudodifferential operator, elliptic in the calculus of van Erp-Yuncken, is hypoelliptic and Fredholm. Under some geometric conditions, we compute its Fredholm index by means of operator -theory. These results extend the work of Baum-van Erp (Acta Mathematica '2014) for co-oriented contact manifolds to a methodology for solving this index problem geometrically on Carnot manifolds. Under the assumption that the Carnot manifold is regular, i.e. has isomorphic osculating Lie algebras in all fibres, and admits a flat coadjoint orbit, the methodology derived from Baum-van Erp's work is developed in full detail. In this case, we develope -theoretical dualities computing the Fredholm index by means of geometric -homology a la Baum-Douglas. The duality involves a Hilbert space bundle of flat orbit representations. Explicit solutions to the index problem for Toeplitz operators and operators of the form "" are computed in geometric -homology, extending results of Boutet de Monvel and Baum-van Erp, respectively, from co-oriented contact manifolds to regular polycontact manifolds.
Paper Structure (59 sections, 168 theorems, 712 equations)

This paper contains 59 sections, 168 theorems, 712 equations.

Table of Contents

  1. Introduction
  2. Background and context
  3. Main results
  4. Two examples of index theorems
  5. Summary of contents
  6. Acknowledgements
  7. Simply connected nilpotent Lie groups
  8. Representation theory
  9. The fine stratification of the spectrum
  10. Flat orbits
  11. Automorphisms and their action on $\hat{\mathsf{G}}$
  12. The continuous trace algebra structure of the ideal of flat orbits
  13. Examples of graded nilpotent Lie groups
  14. Step 1 graded nilpotent Lie groups
  15. Step 2 graded nilpotent Lie groups
  16. ...and 44 more sections

Key Result

Theorem 1

The index problem for an $H$-elliptic $[D]\in K_*(X)$ is solved by

Theorems & Definitions (404)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Remark 2.1
  • Theorem 4
  • Corollary 5
  • Remark 2.2
  • Remark 2.3
  • Theorem 6
  • Remark 2.4
  • ...and 394 more